Free Tools for Rational Education

Free Study Materials for the Casualty Actuarial Society (CAS) Exam 5B 

(Old Exam 6)

Section 15

Uses of Actuarial Triangles for Claims and Claim Counts to Evaluate an Insurer's Situation: Practice Questions and Solutions

G. Stolyarov II
July 18, 2010 - Republished July 11, 2014
Recommend this page.

A sample image

This section is part of Mr. Stolyarov's Free Study Materials for the CAS Exam 5B.

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Estimating Unpaid Claims Using Basic Techniques, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5B (Old Exam 6). Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009. Chapter 6, pp. 70-77.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S6-15-1.

(a) Identify four factors that could change an insurer's ratio of closed claims to reported claims. (See Friedland, p. 71).

(b) If an insurer makes a determined effort to close claims more quickly, how would this affect the mix of open claims at a given time? Explain. (See Friedland, p. 72).

Solution S6-15-1.

(a) The following factors could change an insurer's ratio of closed claims to reported claims (Friedland, p. 71):

1. Natural disasters and other events that interrupt an insurer's systems and lead to delays in closing claims
2. "Change in the guidelines for the establishment of a claim"
3. "Delegation of a higher limit for settlement of claims to a TPA" - a third-party administrator
4. "Introduction of a new call center to handle claims"
5. "Decrease in the statute of limitations, which often accompanies major tort reform"
6. "Restructuring of the claim field offices" - e.g., acquisition of new offices or mergers of existing offices

Any four of the above suffice as an answer. Other valid answers may also be possible.

(b) If an insurer makes a determined effort to close claims more quickly, this will most likely be reflected on settlement patterns for simpler, less expensive claims, where the payout is easier to determine. Larger claims, particularly liability claims involving bodily injury, often take a long time to settle because of circumstances beyond the insurer's control, such as the workings of the court system. Faster settlement of smaller claims would likely change the mix of open claims to one comprised of a greater proportion of larger, more complex claims.

Problem S6-15-2. Various kinds of development triangles are used in analyzing claims in insurance. For each of the following values, name the kinds of triangles that could be used to calculate the value and give the formula involved in the calculation. In your answers, assume that you only have access to reported claim and claim count triangles, paid claim triangles, and closed claim count triangles. (See Friedland, p. 72.)

(a) Average reported claim
(b) Average paid claim
(c) Average case outstanding

Solution S6-15-2.

(a) Average reported claim = (Reported claims)/(Claim count). Use the reported claim triangle and reported claim count triangle.

(b) Average paid claim = (Paid claims)/(Closed claim count). Use the paid claim triangle and the closed claim count triangle.

(c) Average case outstanding = (Reported claims - Paid claims)/(Reported claim count - Closed claim count). Use all four triangles - for reported claims and claim counts, paid claims, and closed claim counts.

Problem S6-15-3. You have the following information for Insurer Q for Accident Year (AY) 2050, as of December 31, 2050:

Reported claims: $314,000
Paid claims: $214,000
Reported claim count: 646
Closed claim count: 441

What is the average case outstanding for AY 2050, as of December 31, 2050?

Solution S6-15-3. We use the formula Average case outstanding = (Reported claims - Paid claims)/(Reported claim count - Closed claim count) = (314000 - 214000)/(646-441) = 487.804878 = $487.80.

Problem S6-15-4. Friedland, on pp. 73-74, discusses a "mismatch" that occurs in average paid claim triangles. Discuss why the mismatch occurs and what it implies.

Solution S6-15-4. The mismatch can be identified by examining the formula Average paid claim = (Paid claims)/(Closed claim count). Paid claims can apply both to claims that are closed and claims that are open but on which partial payments have been made. However, the closed claim count, by definition, only identifies claims that have been fully closed. Thus, the average paid claim formula mistakenly matches partial payments on open claims to closed claims. This might lead to an overestimate of the true average payment on closed claims.

Problem S6-15-5.

(a) Suppose you are examining the average case outstanding triangle of an insurer with a stable book of business (including a stable mix of business and the same policy offerings from year to year). As you move down a particular column, representing average case outstanding at a particular claim age for multiple accident years, what do you expect to observe with regard to average case outstanding trends? (See Friedland, p. 75.)

(b) With regard to an insurer's definition of claim counts, discuss one matter that the actuary examining the insurer's book of business must clarify in order to interpret the data properly. (See Friedland, p. 73.)

Solution S6-15-5.

(a) Average case outstanding for a company with a stable book of business should be expected to change per year by the percentage of general annual inflation. (See Friedland, p. 75.)

(b) With regard to an insurer's definition of claim counts, the actuary must clarify whether or not the definition includes claims closed with no payment (CNP). If CNP claims are included, the average paid claim amount would be systematically lower than if CNP claims were not included, since the denominator is inflated without the numerator being inflated. Moreover, it is important to ascertain whether the insurer's treatment of CNP claims changed during the timeframe under observation. (See Friedland, p. 73.)

Gennady Stolyarov II (G. Stolyarov II) is an actuary, science-fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress. 

In December 2013, Mr. Stolyarov published Death is Wrong, an ambitious children’s book on life extension illustrated by his wife Wendy. Death is Wrong can be found on Amazon in paperback and Kindle formats.

Mr. Stolyarov has contributed articles to the Institute for Ethics and Emerging Technologies (IEET), The Wave Chronicle, Le Quebecois Libre, Brighter Brains Institute, Immortal Life, Enter Stage RightRebirth of Reason, The Liberal Institute, and the Ludwig von Mises Institute. Mr. Stolyarov also published his articles on Associated Content (subsequently the Yahoo! Contributor Network) from 2007 until its closure in 2014, in an effort to assist the spread of rational ideas. He held the highest Clout Level (10) possible on the Yahoo! Contributor Network and was one of its Page View Millionaires, with over 3.1 million views. 

Mr. Stolyarov holds the professional insurance designations of Associate of the Society of Actuaries (ASA), Associate of the Casualty Actuarial Society (ACAS), Member of the American Academy of Actuaries (MAAA), Chartered Property Casualty Underwriter (CPCU), Associate in Reinsurance (ARe), Associate in Regulation and Compliance (ARC), Associate in Personal Insurance (API), Associate in Insurance Services (AIS), Accredited Insurance Examiner (AIE), and Associate in Insurance Accounting and Finance (AIAF).

Mr. Stolyarov has written a science fiction novel, Eden against the Colossus, a philosophical treatise, A Rational Cosmology,  a play, Implied Consent, and a free self-help treatise, The Best Self-Help is Free. You can watch his YouTube Videos. Mr. Stolyarov can be contacted at

Recommend this page.

This TRA feature has been edited in accordance with TRA’s Statement of Policy.

Click here to return to the index of Free Tools for Rational Education.

Learn about Mr. Stolyarov's novel, Eden against the Colossus, here.

Read Mr. Stolyarov's comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, here.

Read Mr. Stolyarov's four-act play, Implied Consent, a futuristic intellectual drama on the sanctity of human life, here.